Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations
1.1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00040505/resource/name/S0008414X00040505_eqn01.gif?pub-status=live)
where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.